Remark 9.4.2.10. Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-compactly generated, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. Then the restriction functor
\[ \operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{c}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{< \kappa }, \operatorname{\mathcal{D}}_{< \kappa } ) \quad \quad F \mapsto F|_{ \operatorname{\mathcal{C}}_{< \kappa } } \]
is an equivalence of $\infty $-categories. Here $\operatorname{\mathcal{C}}_{< \kappa }$ and $\operatorname{\mathcal{D}}_{< \kappa }$ denote the full subcategories spanned by the $(\kappa ,\lambda )$-compact objects of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively. See Proposition 9.4.1.11.